In this chapter you will learn more about whole numbers, and you will strengthen your skills to do calculations and to solve problems.

1.1 Properties of whole numbers 3

1.2 Calculations with whole numbers 7

1.3 Multiples, factors and prime factors 18

1.4 Solving problems 24

1 Whole numbers

1.1 Properties of whole numbers

the commutative property of addition and multiplication

1. Which of the following calculations would you choose to calculate the number of yellow beads in this pattern? Do not do any calculations now, just make a choice.

(a) 7 + 7 + 7 + 7 + 7

(b) 10 + 10 + 10 + 10 + 10 + 10 + 10

(c) 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5

(d) 5 + 5 + 5 + 5 + 5 + 5 + 5

(e) 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7

(f) 10 + 10 + 10 + 10 + 10

My choice:

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2. (a) How many red beads are there in the pattern, and how many yellow beads?

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(b) How many beads are there in the pattern in total?

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3. (a) Which expression describes what you did to calculate the total number of beads:

70 + 50 or 50 + 70?

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(b) Does it make a difference?

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(c) Which expression describes what you did to calculate the number of red beads:

7 \times 10 or 10 \times 7?

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(d) Does it make a difference?

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4. Calculate each of the following:

5 \times 8

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10 \times 8

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12 \times 8

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8 \times 12

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6 \times 8

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3 \times 7

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6 \times 7

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7 \times 6

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the associative property of addition and multiplication

Lebogang and Nathi both have to calculate 25 \times 24.

Lebogang calculates 25 \times 4 and then multiplies by 6.

Nathi calculates 25 \times 6 and then multiplies by 4.

1. Will they get the same answer or not?

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2. Do the following calculations. Do not use a calculator now.

(a) 4 + 7 + 5 + 6

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(b) 7 + 6 + 5 + 4

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(c) 6 + 5 + 7 + 4

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(d) 7 + 5 + 4 + 6

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3. (a) Is addition associative?

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(b) Illustrate your answer with an example.

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4. Find the value of each expression by working in the easiest possible way.

(a) 2 \times 17 \times 5

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(b) 4 \times 7 \times 5

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(c) 75 + 37 + 25

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(d) 60 + 87 + 40 + 13

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5. What must you add to each of the following numbers to get 100?

82 44 56 78 24 89 77

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6. What must you multiply each of these numbers by to get 1 000?

250 125 25 500 200 50

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7. Calculate each of the following. Note that you can make the work very easy by being smart in deciding how to group the operations.

(a) 82 + 54 + 18 + 46 + 237 (b) 24 + 89 + 44 + 76 + 56 + 11

(c) 25 \times (86 \times 4) (d) 32 \times 125

more conventions and the distributive property

The distributive property is a useful property because it allows us to do this:

 

Both answers are 18. Notice that we have to use brackets in the first example to show that the addition operation must be done first. Otherwise, we would have done the multiplication first. For example, the expression 3 \times 2 + 4 means "multiply 3 by 2; then add 4". It does not mean "add 2 and 4; then multiply by 3".

The expression 4 + 3 \times 2 also means "multiply 3 by 2; then add 4".

The distributive property can be used to break up a difficult multiplication into smaller parts. For example, it can be used to make it easier to calculate 6 \times 204:

6 \times 204 can be rewritten as 6 \times (200 + 4) (Remember the brackets!)

= 6 \times 200 + 6 \times 4

= 1 200 + 24

= 1 224

Multiplication can also be distributed over subtraction, for example to calculate 7 \times 96:

7 \times 96 = 7 \times (100 - 4)

= 7 \times 100 - 7 \times 4

= 700 - 28

= 672

1. Here are some calculations with answers. Rewrite them with brackets to make all the answers correct.

(a) 8 + 6 \times 5 = 70 (b) 8 + 6 \times 5 = 38

(c) 5 + 8 \times 6 - 2 = 52 (d) 5 + 8 \times 6 - 2 = 76

(e) 5 + 8 \times 6 - 2 = 51 (f) 5 + 8 \times 6 - 2 = 37

2. Calculate the following:

(a) 100 \times (10 + 7) (b) 100 \times 10 + 100 \times 7

(c) 100 \times (10 - 7) (d) 100 \times 10 - 100 \times 7

3. Complete the table.

\times	8	5	4	9	7	3	6	2	10	11	12
7	56	35	28	63	49	21	42	14	70	77	84
3	24	15	12	27	21	9	18	6	30	33	36
9	72	45	36	81	63	27	54	18	90	99	108
5	40	25	20	45	35	15	30	10	50	55	60
8	64	40	32	72	56	24	48	16	80	88	96
6	48	30	24	54	42	18	36	12	60	66	72
4	32	20	16	36	28	12	24	8	40	44	48
2	16	10	8	18	14	6	12	4	20	22	24
10	80	50	40	90	70	30	60	20	100	110	120
12	96	60	48	108	84	36	72	24	120	132	144
11	88	55	44	99	77	33	66	22	110	121	132

4. Use the various mathematical conventions for numerical expressions to make these calculations easier. Show all your working.

(a) 18 \times 50 (b) 125 \times 28 (c) 39 \times 220

(d) 443 + 2 100 + 557 (e) 318 + 650 + 322 (f) 522 + 3 003 + 78

1.2 Calculations with whole numbers

estimating, approximating and rounding

1. Try to give answers that you trust to these questions, without doing any calculations with the given numbers.

(a) Is 8 \times 117 more than 2 000 or less than 2 000?

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than 2 000

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(b) Is 27 \times 88 more than 3 000 or less than 3 000?

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than 3 000

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(c) Is 18 \times 117 more than 3 000 or less than 3 000?

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than 3 000

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(d) Is 47 \times 79 more than 3 000 or less than 3 000?

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than 3 000

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What you have done when you tried to give answers to questions 1(a) to (d), is called estimation. To estimate is to try to get close to an answer without

actually doing the required calculations with the given numbers.

2. Look at question 1 again.

(a) The numbers 1 000, 2 000, 3 000, 4 000, 5 000, 6 000, 7 000, 8 000, 9 000 and 10 000 are all multiples of a thousand. In each case, write down the multiple of 1 000 that you think is closest to the answer. Write it on the short dotted line. The numbers you write down are called estimates.

(b) In some cases you may think that you may achieve a better estimate by adding 500 to your estimate, or subtracting 500 from it. If so, you may add or subtract 500.

(c) If you wish, you may write what you believe is an even better estimate by adding or subtracting some hundreds.

3. (a) Use a calculator to find the exact answers for the calculations in question 1, or look up the answers in one of the tables on page 2.

Calculate the error in your last approximation of each of the answers in question 1.

(b) What was your smallest error?

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4. Think again about what you did in question 2. In 2(a) you tried to approximate the answers to the nearest 1 000. In 2(c) you tried to approximate the answers to the nearest 100. Describe what you tried to achieve in question 2(b). Sample answer:

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5. Estimate the answers for each of the following products and sums. Try to approximate the answers for the products to the nearest thousand, and for the sums to the nearest hundred. Use the first line in each question to do this.

(a) 84 \times 178

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(b) 677 + 638

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(c) 124 \times 93

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(d) 885 + 473

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(e) 79 \times 84

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(f) 921 + 367

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(g) 56 \times 348

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(h) 764 + 829

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6. Use a calculator to find the exact answers for the calculations in question 5, or look up the answers in the tables on page 2. Calculate the error in each of your approximations. Use the second line in each question to do this.

Calculating with "easy" numbers that are close to given numbers is a good way to obtain approximate answers, for example:

• • To approximate 764 + 829 one may calculate 800 + 800 to get the approximate answer 1 600, with an error of 7.
• • To approximate 84 \times 178 one may calculate 80 \times 200 to get the approximate answer 16 000, with an error of 1 048.

7. Calculate with "easy" numbers close to the given numbers to produce approximate answers for each product below. Do not use a calculator. When you have made your approximations, look up the precise answers in the top table on page 2.

(a) 78 \times 46

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(b) 67 \times 88

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(c) 34 \times 276

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(d) 78 \times 178

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rounding off and compensating

1. (a) Approximate the answer for 386 + 3 435, by rounding both numbers off to the nearest hundred, and adding the rounded numbers.

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(b) Because you rounded 386 up to 400, you introduced an error of 14 in your approximate answer. What error did you introduce by rounding 3 435 down to 3 400?

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(c) What was the combined (total) error introduced by rounding both numbers off before calculating?

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(d) Use your knowledge of the total error to correct your approximate answer, so that you have the correct answer for 386 + 3 435.

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What you have done in question 1 to find the correct answer for 386 + 3 435 is called rounding off and compensating. By rounding the numbers off you introduced errors. You then compensated for the errors by making adjustments to your answer.

2. Round off and compensate to calculate each of the following accurately:

(a) 473 + 638 Sample answer: (b) 677 + 921 Sample answer:

Subtraction can also be done in this way. For example, to calculate R5 362 - R2 687, you may round R2 687 up to R3 000. The calculation can proceed as follows:

• • Rounding R2 687 up to R3 000 can be done in two steps: 2 687 + 13 = 2 700, and 2 700 + 300 = 3 000. In total, 313 is added.
• • 313 can now be added to 5 362 too: R5 362 + 313 = 5 675.
• • Instead of calculating R5 362 - R2 687, which is a bit difficult, you may calculate R5 675 - R3 000. This is easy: R5 675 - R3 000 = R2 675.

This means that R5 362 - R2 687 = R2 675, because

R5 362 - R2 687 = (R5 362 + R313) - (R2 687 + R313).

adding numbers in parts written in columns

Numbers can be added by thinking of their parts as we say the numbers.

For example, we say 4 994 as four thousand nine hundred and ninety-four. This can be written in expanded notation as 4 000 + 900 + 90 + 4.

Similarly, we can think of 31 837 as 30 000 + 1 000 + 800 + 30 + 7.

31 837 + 4 994 can be calculated by working with the various kindsof parts separately. To make this easy, the numbers can be written below each other so that the units are below the units, the tens below the tens and so on, as shown on the right.

We write only this:	In your mind you can see this:
31 837	30 000	1 000	800	30	7
4 994		4 000	900	90	4

The numbers in each column can be added to get a new set of numbers.

31 837	30 000	1 000	800	30	7
4 994		4 000	900	90	4
11					11
120				120
1 700			1 700
5 000		5 000
30 000	30 000
36 831	It is easy to add the new set of numbers to get the answer.

The work may start with the 10 000s or any other parts. Starting with the units as shown above makes it possible to do more of the work mentally, and write less, as shown below.

31 837	To achieve this, only the units digit 1 of the 11 is written in the first step. The 10 of the 11 is remembered and added to the 30 and 90 of the tens column, to get 130.
4 994
36 831

We say the 10 is carried from the units column to the tens column. The same is done when the tens parts are added to get 130: only the digit "3" is written (in the tens column, so it means 30), and the 100 is carried to the next step.

1. Calculate each of the following without using a calculator:

(a) 4 638 + 2 667 (b) 748 + 7 246

2. Impilo Enterprises plans a new computerised training facility in their existing building. The training manager has to keep the total expenditure budget under R1 million. This is what she has written so far:

Architects and builders Painting and carpeting Security doors and blinds Data projector 25 new secretary chairs 24 desks for work stations 1 desk for presenter 25 new computers 12 colour laser printers

R102 700 R 42 600 R 52 000 R 4 800 R 50 400 R123 000 R 28 000 R300 000 R 38 980

Work out the total cost of all the items the training manager has budgeted for.

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3. Calculate each of the following without using a calculator:

(a) 7 828 + 6 284 (b) 7 826 + 888 + 367

(c) 657 + 32 890 + 6 542 (d) 6 666 + 3 333 + 1

methods of subtraction

There are many ways to find the difference between two numbers. For example, to find the difference between 267 and 859 one may think of the numbers as they may be written on a number line.

 

We may think of the distance between 267 and 859 as three steps: from 267 to 300, from 300 to 800, and from 800 to 859. How big are each of these three steps?

 

The above shows that 859 - 267 is 33 + 500 + 59.

1. Calculate 33 + 500 + 59 to find the answer for 859 - 267.

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2. Calculate each of the following. You may think of working out the distance between the two numbers as shown above, or use any other method you prefer. Do not use a calculator now.

(a) 823 - 456 (b) 1 714 - 829

(c) 3 045 - 2 572 (d) 5 131 - 367

You can use the tables of sums on page 2 to check your answers for question 2.

Like addition, subtraction can also be done by working with the different parts in which we say numbers. For example, 8 764 - 2 352 can be calculated as follows:

8 thousand - 2 thousand = 6 thousand

7 hundred - 3 hundred = 4 hundred

6 tens - 5 tens = 1 ten

4 units - 2 units = 2 units

So, 8 764 - 2 352 = 6 412

Subtraction by parts is more difficult in some cases, for example 6 213 - 2 758:

6 000 - 2 000 = 4 000. This step is easy, but the following steps cause problems:

200 - 700 = ?

10 - 50 = ?

3 - 8 = ?

Fortunately, the parts and sequence of work may be rearranged to overcome these problems, as shown below:

instead of			we may do
3 - 8 =	?		13 - 8 =		"borrow" 10 from below
10 - 50 =	?		100 - 50 =		"borrow" 100 from below
200 - 700 =	?		1 100 - 700 =		"borrow" 1 000 from below
6 000 - 2 000 =	?		5 000 - 2 000 =

This reasoning can also be set out in columns:

instead of					we may do					but write only this
6 000	200	10	3		5 000	1 100	100	13		6	2	1	3
2 000	700	50	8		2 000	700	50	8		2	7	5	8
					3 000	400	50	5		3	4	5	5

3. (a) Complete the above calculations and find the answer for 6 213 - 2 758.

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(b) Use the borrowing technique to calculate 823 - 376 and 6 431 - 4 968.

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4. Check your answers in question 3(b) by doing addition.

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376 4 968

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With some practice, you can learn to subtract using borrowing without writing all the steps. It is convenient to work in columns, as shown on the right for calculating 6 213 - 2 758.

In fact, by doing more work mentally, you may learn to save more paper by writing even less as shown below.

Do not use a calculator when you do question 5, because the purpose of this work is for you to come to understand methods of subtraction. What you will learn here will later help you to understand algebra better.

5. Calculate each of the following:

(a) 7 342 - 3 877 (b) 8 653 - 1 856 (c) 5 671 - 4 528

You may use a calculator to do questions 6 and 7.

6. Estimate the difference between the two car prices in each case to the nearest R1 000 or closer. Then calculate the difference.

(a) R102 365 and R98 128 (b) R63 378 and R96 889

7. First estimate the answers to the nearest 100 000 or 10 000 or 1 000. Then calculate.

(a) 238 769 -141 453 (b) 856 333 - 739 878 (c) 65 244 - 39 427

a method of multiplication

7 \times 4 598 can be calculated in parts, as shown here:

7 \times 4 000 = 28 000

7 \times 500 = 3 500

7 \times 90 = 630

7 \times 8 = 56

The four partial products can now be added to get the answer, which is 32 186. It is convenient to write the work in vertical columns for units, tens, hundreds and so on, as shown on the right.

The answer can be produced with less writing, by "carrying" parts of the partial answers to the next column, when working from right to left in the columns.

Only the 6 of the product 7 \times 8 is written down instead of 56. The 50 is kept in mind, and added to the 630 obtained when 7 \times 90 is calculated in the next step.

1. Calculate each of the following. Do not use a calculator now.

(a) 27 \times 649 (b) 75 \times 1 756 (c) 348 \times 93

2. Use your calculator to check your answers for question 1. Redo the questions for which you had the wrong answers.

3. Calculate each of the following. Do not use a calculator now.

(a) 67 \times 276 (b) 84 \times 178

4. Use the product table on page 2 or a calculator to check your answers for question 3. Redo the questions for which you had the wrong answers.

long division

1. The municipal head gardener wants to buy young trees to plant along the main street of the town. The young trees cost R27 each, and an amount of R9 400 has been budgeted for trees. He needs 324 trees. Do you think he has enough money?

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2. (a) How much will 300 trees cost?

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(b) How much money will be left if 300 trees are bought?

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(c) How much money will be left if 20 more trees are bought?

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The municipal gardener wants to work out exactly how many trees, at R27 each, he can buy with the budgeted amount of R9 400. His thinking and writing are described below.

Step 1

What he writes:					What he thinks:
27	9400		Iwant to find out how many chunks of 27 there are in 9 400.

Step 2

What he writes:					What he thinks:
	300		Ithink there are at least 300 chunks of 27 in 9 400.
27	9400
	8100		300 \times 27 = 8 100. I need to know how much is left over.
	1300		Iwant to find out how many chunks of 27 there are in 1 300.

Step 3 (He has to rub out the one "0" of the 300 on top, to make space.)

What he writes:					What he thinks:
	340		Ithink there are at least 40 chunks of 27 in 1 300.
27	9400
	8100
	1300
	1080		40 \times 27 = 1 080. I need to know how much is left over.
	220		Iwant to find out how many chunks of 27 there are in 220.Perhaps I can buy some extra trees.

Step 4 (He rubs out another "0".)

What he writes:					What he thinks:
	348		Ithink there are at least 8 chunks of 27 in 220.
27	9400
	8100
	1300
	1080
	220
	216		8\times 27 = 216
	4		So, I can buy 348 young trees and will have R4 left.

Do not use a calculator to do questions 3 and 4. The purpose of this work is for you to develop a good understanding of how division can be done. Check all your answers by doing multiplication.

3. (a) Graham bought 64 goats, all at the same price. He paid R5 440 in total. What was the price for each goat? Your first step can be to work out how much he would have paid if he paid R10 per goat, but you can start with a bigger step if you wish.

(b) Mary has R2 850 and she wants to buy candles for her sister's wedding reception. The candles cost R48 each. How many candles can she buy?

4. Calculate each of the following, without using a calculator:

(a) 7 234 \div 48 (b) 3 267 \div 24

(c) 9 500 \div 364 (d) 8 347 \div 24

1.3 Multiples, factors and prime factors

multiples and factors

1. The numbers 6; 12; 18; 24; ... are multiples of 6.

The numbers 7; 14; 21; 28; ... are multiples of 7.

(a) What is the 100th number in each sequence above?

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(b) Is 198 a number in the first sequence?

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(c) Is 175 a number in the second sequence?

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Of which numbers is 20 a multiple?

20 = 1 \times 20 = 2 \times 10 = 4 \times 5 = 5\times 4 = 10 \times 2 = 20 \times 1

Factors come in pairs. The following pairs are factors of 20:

 

2. A rectangle has an area of 30 cm. What are the possible lengths of the sides of the rectangle in centimetres if the lengths of the sides are natural numbers?

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3. Are 4; 8; 12 and 16 factors of 48? Simon says that all multiples of 4 smaller than 48 are factors of 48. Is he right?

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4. We have defined factors in terms of the product of two numbers. What happens if we have a product of three or more numbers, for example 210 = 2 \times 3 \times 5 \times 7?

(a) Explain why 2; 3; 5 and 7 are factors of 210.

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(b) Are 2 \times 3; 3 \times 5; 5 \times 7; 2 \times 5 and 2 \times 7 factors of 210?

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(c) Are 2 \times 3 \times 5; 3 \times 5 \times 7 and 2 \times 5 \times 7 factors of 210?

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5. Is 20 a factor of 60? What factors of 20 are also factors of 60?

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prime numbers and composite numbers

1. Express each of the following numbers as a product of as many factors as possible, including repeated factors. Do not use 1 as a factor.

(a) 66 (b) 67

(c) 68 (d) 69

(e) 70 (f) 71

(g) 72 (h) 73

2. Which of the numbers in question 1 cannot be expressed as a product of two whole numbers, except as the product 1\times the number itself?

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3. Which of the numbers in question 1 are prime?

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4. Are the statements below true or false? If you answer "false", explain why.

(a) All prime numbers are odd numbers.

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(b) All composite numbers are even numbers.

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(c) 1 is a prime number.

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(d) If a natural number is not prime, then it is composite.

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(e) 2 is a composite number.

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(f) 785 is a prime number.

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(g) A prime number can only end in 1; 3; 7 or 9.

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(h) Every composite number is divisible by at least one prime number.

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5. We can find out whether a given number is prime by systematically checking whether the primes 2; 3; 5; 7; 11; 13; … are factors of the given number or not.

To find possible factors of 131, we need to consider only the primes 2; 3; 5; 7 and 11. Why not 13; 17; 19; …?

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6. Determine whether the following numbers are prime or composite. If the number is composite, write down at least two factors of the number (besides 1 and the number itself).

(a) 221 (b) 713

prime factorisation

To find all the factors of a number you can write the number as the product of prime factors, first by writing it as the product of two convenient (composite) factors and then by splitting these factors into smaller factors until all factors are prime. Then you take all the possible combinations of the products of the prime factors.

Example: Find the factors of 84.

Write 84 as the product of prime factors by starting with different known factors:

84 = 4 \times 21 or 84 = 7 \times 12 or 84 = 2 \times 42

= 2 \times 2 \times 3 \times 7 = 7 \times 3 \times 4 = 2 \times 6 \times 7

= 7 \times 3 \times 2 \times 2 = 2 \times 2 \times 3 \times 7

A more systematic way of finding the prime factors of a number would be to start with the prime numbers and try the consecutive prime numbers 2; 3; 5; 7; ... as possible factors. The work may be set out as shown below.

2	1 430		3	2 457
5	715		3	819
11	143		3	273
13	13		7	91
	1		13	13
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1 430 = 2 \times 5 \times 11 \times 13 2 457 = 3 \times 3 \times 3 \times 7 \times 13

We can use exponents to write the products of prime factors more compactly as products of powers of prime factors.

2 457 = 3 \times 3 \times 3 \times 7 \times 13 = 33 \times 7 \times 13

72 = 2 \times 2 \times 2 \times 3 \times 3 = 23 \times 32

1 500 = 2 \times 2 \times 3 \times 5 \times 5 \times 5 = 22 \times 3 \times 53

1. Express the following numbers as the product of powers of primes:

(a) 792 =

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(b) 444 =

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2. Find the prime factors of the numbers below.

2	28			32			124			36			42			345			182
	14

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common multiples and factors

1. Is 4 \times 5 a multiple of 4?

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Is 4 \times 5 a multiple of 5?

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2. Comment on the following statement: The product of numbers is a multiple of each of the numbers in the product.

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We use common multiples when fractions with different denominators are added.

To add + , the common denominator is 3 \times 4, so the sum becomes + .

In the same way, we could use 6 \times 8 = 48 as a common denominator to add + , but

24 is the lowest common multiple (LCM) of 6 and 8.

Prime factorisation makes it easy to find the lowest common multiple or highest common factor.

When we simplify a fraction, we divide the same number into the numerator and the denominator. For the simplest fraction, use the highest common factor (HCF) to divide into both numerator and denominator.

So = =  

Use prime factorisation to determine the LCM and HCF of 32, 48 and 84 in a systematic way: 32 = 2 \times 2 \times 2 \times 2 \times 2 = 25

48 = 2 \times 2 \times 2 \times 2 \times 3 = 24 \times 3

84 = 2 \times 2 \times 3 \times 7 = 22 \times 3 \times 7

The LCM is a multiple, so all of the factors of all the numbers must divide into it.

All of the factors that are present in the three numbers must also be factors of the LCM, even if it is a factor of only one of the numbers. But because it has to be the lowest common multiple, no unnecessary factors are in the LCM.

The highest power of each factor is in the LCM, because then all of the other factors can divide into it. In 32, 48 and 84, the highest power of 2 is 25, the highest power of 3 is 3 and the highest power of 7 is 7.

LCM = 25 \times 3 \times 7 = 672

The HCF is a common factor. Therefore, for a factor to be in the HCF, it must be a factor of all of the numbers. 2 is the only number that appears as a factor of all three numbers. The lowest power of 2 is 22, so the HCF is 22.

3. Determine the LCM and the HCF of the numbers in each case.

(a) 24; 28; 42 (b) 17; 21; 35

(c) 75; 120; 200 (d) 18; 30; 45

investigate prime numbers

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You may use a calculator for this investigation. 1. Find all the prime numbers between 110 and 130. 2. Find all the prime numbers between 210 and 230. 3. Find the biggest prime number smaller than 1 000.

1.4 Solving problems

rate and ratio

You may use a calculator for the work in this section.

1. Tree plantations in the Western Cape are to be cut down in favour of natural vegetation. There are roughly 3 000 000 trees on plantations in the area and it is possible to cut them down at a rate of 15 000 trees per day with the labour available. How many working days will it take before all the trees will be cut down?

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2. A car travels a distance of 180 km in 2 hours on a straight road. How many kilometres can it travel in 3 hours at the same speed?

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3. Thobeka wants to order a book that costs $56,67. The rand-dollar exchange rate is R7,90 to a dollar. What is the price of the book in rands?

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4. In pattern A below, there are 5 red beads for every 4 yellow beads.

 

Describe patterns B and C in the same way.

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5. Complete the table to show how many screws are produced by two machines in different periods of time.

Number of hours	1	2	3	5	8
Number of screws at machine A	1 800	3 600	5 400	9 000	14 400
Number of screws at machine B	2 700	5 400	8 100	13 500	21 600

(a) How much faster is machine B than machine A?

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(b) How many screws will machine B produce in the same time that it takes machine A to make 100 screws?

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The patterns in question 4 can be described like this: In pattern A, the ratio of yellow beads to red beads is 4 to 5. This is written as 4 : 5. In pattern B, the ratio between yellow beads and red beads is 3 : 6. In pattern C the ratio is 2 : 7.

In question 5, machine A produces 2 screws for every 3 screws that machine B produces. This can be described by saying that the ratio between the production speeds of machines A and B is 2 : 3.

6. Nathi, Paul and Tim worked in Mr Setati's garden. Nathi worked for 5 hours, Paul for 4 hours and Tim for 3 hours. Mr Setati gave the boys R600 for their work. How should they divide the R600 among the three of them?

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A ratio is a comparison of two (or more) quantities.

The number of hours that Nathi, Paul and Tim worked are in the ratio 5 : 4 : 3. To be fair, the money should also be shared in that ratio. That means that Nathi should receive 5 parts, Paul 4 parts and Tim 3 parts of the money. There were 12 parts, which means Nathi should

receive of the total amount, Paul should get and Tim should get .

7. Ntabi uses 3 packets of jelly to make a pudding for 8 people. How many packets of jelly does she need to make a pudding for 16 people? And for 12 people?

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8. Which rectangle is more like a square: a 3 \times 5 rectangle or a 6 \times 8 rectangle? Explain.

A 6 \times 8 rectangle, because is closer to 1 than is.

9. (a) Increase 56 in the ratio 2 : 3.

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(b) Decrease 72 in the ratio 4 : 3.

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10. (a) Divide 840 in the ratio 3 : 4.

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(b) Divide 360 in the ratio 1 : 2 : 3.

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11. Some data about the performance of different athletes during a walking event is given below. Investigate the data to find out who walks fastest and who walks slowest. Arrange the athletes from the fastest walker to the slowest walker.

(a) First make estimates to do the investigation.

(b) Then use your calculator to do the investigation.

Athlete	A	B	C	D	E	F
Distance walked in m	2 480	4 283	3 729	6 209	3 112	5 638
Time taken in minutes	17	43	28	53	24	45

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profit, loss, discount and interest

1. (a) How much is 1 eighth of R800?

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(b) How much is 1 hundredth of R800?

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(b) How much is 7 hundredths of R800?

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Rashid is a furniture dealer. He buys a couch for R2 420. He displays the couch in his showroom with the price marked as R3 200. Rashid offers a discount of R320 to customers who pay cash.

2. (a) What is the cost price of the couch in Rashid's furniture shop?

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(b) What is the marked price?

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(c) What is the selling price for a customer who pays cash?

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(d) How much is 10 hundredths of R3 200?

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The discount on an article is always less than the marked price of the article. In fact, it is only a fraction of the marked price. The discount of R320 that Rashid offers on the couch is 10 hundredths of the marked price.

Another word for hundredths is percentage, and the symbol for percentage is %. So we can say that Rashid offers a discount of 10%.

A percentage is a number of hundredths.

18% is 18 hundredths, and 25% is 25 hundredths.

A discount of 6% on an article can be calculated in two steps:

Step 1: Calculate 1 hundredth of the marked price (divide by 100).

Step 2: Calculate 6 hundredths of the marked price (multiply by 6).

3. Calculate a discount of 6% on each of the following marked prices of articles:

(a) R3 600 (b) R9 360

4. (a) How much is 1 hundredth of R700?

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(b) A customer pays cash for a coat marked at R700. He is given R63 discount. How many hundredths of R700 is this?

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(c) What is the percentage discount?

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5. A client buys a blouse marked at R300 and she is given R36 discount for paying cash. Work as in question 4 to determine what percentage discount she was given.

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You may use a calculator to do questions 6, 7 and 8.

6. A dealer buys an article for R7 500 and makes the price 30% higher. The article is sold at a 20% discount.

(a) What is the selling price of the article?

of 7 500 = R2 250 Marked price: R7 500 + R2 250 = R9 750

Selling price: of R9 750 = R7 800

(b) What is the dealer's percentage profit?

R7 800 - R7 500 = R300 \times = 4%

When a person borrows money from a bank or some other institution, he or she normally has to pay for the use of the money. This is called interest.

7. Sam borrows R7 000 from a bank at 14% interest for one year. How much does he have to pay back to the bank at the end of the period?

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8. Jabu invests R5 600 for one year at 8% interest.

(a) What will the value of his investment be at the end of that year?

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(b) At the end of the year Jabu does not withdraw the investment or the interest earned, but reinvests it for another year. How much will it be worth at the end of the second year?

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(c) What will the value of Jabu's investment be after five years?

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In this chapter you will work with whole numbers smaller than 0. These numbers are called negative numbers. The whole numbers larger than 0, 0 itself and the negative whole numbers together are called the integers. Mathematicians have agreed that negative numbers should have certain properties that would make them useful for various purposes. You will learn about these properties and how they make it possible to do calculations with negative numbers.

2.1 What is beyond 0? 31

2.2 Adding and subtracting with integers 35

2.3 Multiplying and dividing with integers 40

2.4 Squares, cubes and roots with integers 47

Do what you can.

5 - 0 = ?	5 - 7 = ?	5 + 5 = ?
5 - 1 = ?	5 - 6 = ?	5 + 4 = ?
5 - 2 = ?	5 - 5 = ?	5 + 3 = ?
5 - 3 = ?	5 - 4 = ?	5 + 2 = ?
5 - 4 = ?	5 - 3 = ?	5 + 1 = ?
5 - 5 = ?	5 - 2 = ?	5 + 0 = ?
5 - 6 = ?	5 - 1 = ?	5 + ? = ?
5 - 7 = ?	5 - 0 = ?	5 + ? = ?
5 - 8 = ?	5 - ? = ?	5 + ? = ?
5 - 9 = ?	5 -? = ?	5 + ? = ?
5 - 10 = ?	5 - ? = ?	5 + ? = ?

2 Integers

2.1 What is beyond 0?

why people decided to have negative numbers

On the right, you can see how Jimmy prefers to work when doing calculations such as 542 + 253.

He tries to calculate 542 - 253 in a similar way:

500 - 200 = 300

40 - 50 = ?

Jimmy clearly has a problem. He reasons as follows:

Ican subtract 40 from 40; that gives 0. But then there is still 10 that I have to subtract.

He decides to deal with the 10 that he still has to subtract later, and continues:

500 - 200 = 300

40 - 50 = 0, but there is still 10 that I have to subtract.

2 - 3 = 0, but there is still 1 that I have to subtract.

1. (a) What must Jimmy still subtract, and what will his final answer be?

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(b) When Jimmy did another subtraction problem, he ended up with this writing at one stage:

600 and (-)50 and (-)7

What do you think is Jimmy's final answer for this subtraction problem?

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About 500 years ago, some mathematicians proposed that a "negative number" may be used to describe the result in a situation such as in Jimmy's subtraction problem above, where a number is subtracted from a number smaller than itself.

For example, we may say 10 - 20 = (-10)

This proposal was soon accepted by other mathematicians, and it is now used all over the world.

2. Calculate each of the following:

(a) 16 - 20

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(b) 16 - 30

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(c) 16 - 40

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(d) 16 - 60

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(e) 16 - 200

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(f) 5 - 1 000

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3. Some numbers are shown on the lines below. Fill in the missing numbers.

 

 

The following statement is true if the number is 5:

15 - (a certain number) = 10

A few centuries ago, some mathematicians decided they wanted to have numbers that will also make sentences like the following true:

15 + (a certain number) = 10

But to go from 15 to 10 you have to subtract 5.

The number we need to make the sentence 15 + (a certain number) = 10 true must have the following strange property:

If you add this number, it should have the same effect as to subtract 5.

Now the mathematicians of a few centuries ago really wanted to have numbers for which such strange sentences would be true. So they thought:

Let us decide, and agree amongst ourselves, that the number we call negative 5 will have the property that if you add it to another number, the effect will be the same as when you subtract the natural number 5.

This means that the mathematicians agreed that 15 + (-5) is equal to 15 - 5.

Stated differently, instead of adding negative 5 to a number, you may subtract 5.

4. Calculate each of the following:

(a) 500 + (-300)

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(b) 100 + (-20) + (-40)

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(c) 500 + (-200) + (-100)

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(d) 100 + (-60)

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5. Make a suggestion of what the answer for (-20) + (-40) should be. Give reasons for your suggestion.

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6. Continue the lists of numbers below to complete the table.

(a)	(b)	(c)	(d)	(e)	(f)	(g)
10	100	3	-3	-20	150	0
9	90	6	-6	-18	125	-5
8	80	9	-9	-16	100	-10
7	70	12	-12	-14	75	-15
6	60	15	-15	-12	50	-20
5	50	18	-18	-10	25	-25
4	40	21	-21	-8	0	-30
3	30	24	-24	-6	-25	-35
2	20	27	-27	-4	-50	-40
1	10	30	-30	-2	-75	-45
-1	-10	33	-33	2	-100	-50
-2	-20	36	-36	4	-125	-55
-3	-30	39	-39	6	-150	-60
-4	-40	42	-42	8	-175	-65

The following statement is true if the number is 5:

15 +(a certain number) = 20

What properties should a number have so that it makes the following statement true?

15 - (a certain number) = 20

To go from 15 to 20 you have to add 5. The number we need to make the sentence 15 - (a certain number) = 20 true must have the following property:

If you subtract this number, it should have the same effect as to add 5.

Let us agree that 15 - (-5) is equal to 15 + 5.

Stated differently, instead of subtracting negative 5 from a number, you may add 5.

7. Calculate.

(a) 30 - (-10)

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(b) 30 + 10

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(c) 30 + (-10)

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(d) 30 - 10

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(e) 30 - (-30)

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(f) 30 + 30

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(g) 30 + (-30)

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(h) 30 - 30

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You probably agree that

5 + (-5) = 0 10 + (-10) = 0 and 20 + (-20) = 0

We may say that for each "positive" number there is a corresponding or opposite negative number. Two positive and negative numbers that correspond, for example 3 and (-3), are called additive inverses. They wipe each other out when you add them.

8. Write the additive inverse of each of the following numbers:

(a) 24

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(b) -24

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(c) -103

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(d) 2 348

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The idea of additive inverses may be used to explain why 8 + (-5) is equal to 3:

8 + (-5) = 3 + 5 + (-5) = 3 + 0 = 3

9. Use the idea of additive inverses to explain why each of these statements is true:

(a) 43 + (-30) = 13 (b) 150 + (-80) = 70

STATEMENTS THAT ARE TRUE FOR MANY DIFFERENT NUMBERS

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For how many different pairs of numbers can the following statement be true, if only natural (positive) numbers are allowed? a number + another number = 10 For how many different pairs of numbers can the statement be true if negative numbers are also allowed?

2.2 Adding and subtracting with integers

adding can make less and subtraction can make more

1. Calculate each of the following:

(a) 10 + 4 + (-4)

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(b) 10 + (-4) + 4

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(c) 3 + 8 + (-8)

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(d) 3 + (-8) + 8

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Natural numbers can be arranged in any order to add and subtract them. This is also the case for integers.

2. Calculate each of the following:

(a) 18 + 12

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(b) 12 + 18

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(c) 2 + 4 + 6

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(d) 6 + 4 + 2

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(e) 2 + 6 + 4

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(f) 4 + 2 + 6

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(g) 4 + 6 + 2

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(h) 6 + 2 + 4

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(i) 6 + (-2) + 4

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(j) 4 + 6 + (-2)

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(k) 4 + (-2) + 6

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(l) (-2) + 4 + 6

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(m) 6 + 4 + (-2)

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(n) (-2) + 6 + 4

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(o) (-6) + 4 + 2

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3. Calculate each of the following:

(a) (-5) + 10

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(b) 10 + (-5)

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(c) (-8) + 20

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(d) 20 - 8

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(e) 30 + (-10)

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(f) 30 + (-20)

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(g) 30 + (-30)

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(h) 10 + (-5) + (-3)

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(i) (-5) + 7 + (-3) + 5

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(j) (-5) + 2 + (-7) + 4

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4. In each case, find the number that makes the statement true. Give your answer by writing a closed number sentence.

(a) 20 + (an unknown number) = 50

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(b) 50 + (an unknown number) = 20

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(c) 20 + (an unknown number) = 10

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(d) (an unknown number) + (-25) = 50

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(e) (an unknown number) + (-25) = -50

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5. Use the idea of additive inverses to explain why each of the following statements is true:

(a) 43 + (-50) = -7

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(b) 60 + (-85) = -25

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6. Complete the table as far as you can.

(a)	(b)	(c)
5 - 8 = -3	5 + 8 = -3	8 - 3 = -3
5 - 7 = -3	5 + 7 = -3	7 - 3 = -3
5 - 6 = -3	5 + 6 = -3	6 - 3 = -3
5 - 5 = -3	5 + 5 = -3	5 - 3 = -3
5 - 4 = -3	5 + 4 = -3	4 - 3 = -3
5 - 3 = -3	5 + 3 = -3	3 - 3 = -3
5 - 2 = -3	5 + 2 = -3	2 - 3 = -3
5 - 1 = -3	5 + 1 = -3	1 - 3 = -3
5 - 0 = -3	5 + 0 = -3	0 - 3 = -3
5 - (-1) = -3	5 + (-1) = -3	(-1) - 3 = -3
5 - (-2) = -3	5 + (-2) = -3	(-2) - 3 = -3
5 - (-3) = -3	5 + (-3) = -3	(-3) - 3 = -3
5 - (-4) = -3	5 + (-4) = -3	(-4) - 3 = -3
5 - (-5) = -3	5 + (-5) = -3	(-5) - 3 = -3
5 - (-6) = -3	5 + (-6) = -3	(-6) - 3 = -3

7. Calculate.

(a) 80 + (-60) (b) 500 + (-200) + (-200)

8. (a) Is 100 + (-20) + (-20) = 60, or does it equal something else?

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(b) What do you think (-20) + (-20) is equal to?

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9. Calculate.

(a) 20 - 20

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(b) 50 – 20

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(c) (-20) - (-20)

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(d) (-50) - (-20)

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10. Calculate.

(a) 20 - (-10)

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(b) 100 - (-100)

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(c) 20 + (-10)

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(d) 100 + (-100)

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(e) (-20) - (-10)

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(f) (-100) - (-100)

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(g) (-20) + (-10)

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(h) (-100) + (-100)

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11. Complete the table as far as you can.

(a)	(b)	(c)
5 - (-8) =-3	(-5) + 8 = -3	8 - (-3) = -3
5 - (-7) = -3	(-5) + 7 = -3	7 - (-3) = -3
5 - (-6) = -3	(-5) + 6 = -3	6 - (-3) = -3
5 - (-5) = -3	(-5) + 5 = -3	5 - (-3) = -3
5 - (-4) = -3	(-5) + 4 = -3	4 - (-3) = -3
5 - (-3) = -3	(-5) + 3 = -3	3 - (-3) = -3
5 - (-2) = -3	(-5) + 2 = -3	2 - (-3) = -3
5 - (-1) = -3	(-5) + 1 = -3	1 - (-3) = -3
5 - 0 = -3	(-5) + 0 = -3	0 - (-3) = -3
5 - 1 = -3	(-5) + (-1) = -3	(-1) - (-3) = -3
5 - 2 = -3	(-5) + (-2) = -3	(-2) - (-3) = -3
5 - 3 = -3	(-5) + (-3) = -3	(-3) - (-3) = -3
5 - 4 = -3	(-5) + (-4) = -3	(-4) - (-3) = -3
5 - 5 = -3	(-5) + (-5) = --3	(-5) - (-3) = -3

12. In each case, state whether the statement is true or false and give a numerical example to demonstrate your answer.

(a) Subtracting a positive number from a negative number has the same effect as adding the additive inverse of the positive number.

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(b) Adding a negative number to a positive number has the same effect as adding the additive inverse of the negative number.

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(c) Subtracting a negative number from a positive number has the same effect as subtracting the additive inverse of the negative number.

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(d) Adding a negative number to a positive number has the same effect as subtracting the additive inverse of the negative number.

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(e) Adding a positive number to a negative number has the same effect as adding the additive inverse of the positive number.

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(f) Adding a positive number to a negative number has the same effect as subtracting the additive inverse of the positive number.

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(g) Subtracting a positive number from a negative number has the same effect as subtracting the additive inverse of the positive number.

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(h) Subtracting a negative number from a positive number has the same effect as adding the additive inverse of the negative number.

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comparing integers and solving problems

1. Fill <, > or = into the block to make the relationship between the numbers true:

(a) -103

-99

(b) -699

-701

(c) 30

-30

(d) 10 - 7

-(10 - 7)

(e) -121

-200

(f) 12 - 5

-(12 + 5)

(g) -199

-110

2. At 5 a.m. in Bloemfontein the temperature was -5 °C. At 1 p.m., it was 19 °C. By how many degrees did the temperature rise?

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3. A diver swims 150 m below the surface of the sea. She moves 75 m towards the surface. How far below the surface is she now?

-150 m + 75 m = -75 m

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4. One trench in the ocean is 800 m deep and another is 2 200 m deep. What is the difference in their depths?

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5. An island has a mountain which is 1 200 m high. The surrounding ocean has a depth of 860 m. What is the difference in height?

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6. On a winter's day in Upington the temperature rose by 19 °C. If the minimum temperature was -4 °C, what was the maximum temperature?

-4 °C + 19 °C = 15 °C

2.3 Multiplying and dividing with integers

multiplication with integers

1. Calculate.

(a) -5 + -5 + -5 + -5 + -5 + -5 + -5 + -5 + -5 + -5

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(b) -10 + -10 + -10 + -10 + -10

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(c) -6 + -6 + -6 + -6 + -6 + -6 + -6 + -6

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(d) -8 + -8 + -8 + -8 + -8 + -8

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(e) -20 + -20 + -20 + -20 + -20 + -20 + -20

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2. In each case, show whether you agree (✓) or disagree (✗) with the given statement.

(a) 10 \times (-5) = 50

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(b) 8 \times (-6) = (-8) \times 6

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(c) (-5) \times 10 = 5 \times (-10)

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(d) 6 \times (-8) = -48

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(e) (-5) \times 10 = 10 \times (-5)

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(f) 8 \times (-6) = 48

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(g) 4 \times 12 = -48

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(h) (-4) \times 12 = -48

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3. Is addition of integers commutative? Demonstrate your answer with three different examples.

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4. Calculate.

(a) 20 \times (-10)

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(b) (-5) \times 4

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(c) (-20) \times 10

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(d) 4 \times (-25)

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(e) 29 \times (-20)

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(f) (-29) \times (-2)

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5. Calculate.

(a) 10 \times 50 + 10 \times (-30) (b) 50 + (-30)

(c) 10 \times (50 + (-30)) (d) (-50) + (-30)

(e) 10 \times (-50) + 10 \times (-30) (f) 10 \times ((-50) + (-30))

6. (a) Four numerical expressions are given below. Underline the expressions that you would expect to have the same answers. Do not do the calculations.

14 \times (23 + 58) 23 \times (14 + 58) 14 \times 23 + 14 \times 58 14 \times 23 + 58

(b) What property of operations is demonstrated by the fact that two of the above expressions have the same value?

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7. Consider your answers for question 6.

(a) Does multiplication distribute over addition in the case of integers?

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(b) Illustrate your answer with two examples.

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8. Three numerical expressions are given below. Underline the expressions that you would expect to have the same answers. Do not do the calculations.

10 \times ((-50) - (-30)) 10 \times (-50) - (-30) 10 \times (-50) - 10 \times (-30)

9. Do the three sets of calculations given in question 8.

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Your work in questions 5, 8 and 9 demonstrates that multiplication with a positive number distributes over addition and subtraction of integers. For example:

10 \times (5 + (-3)) = 10 \times 2 = 20 and 10 \times 5 + 10 \times (-3) = 50 + (-30) = 20

10 \times (5 - (-3)) = 10 \times 8 = 80 and 10 \times 5 - 10 \times (-3) = 50 - (-30) = 80

10. Calculate: (-10) \times (5 + (-3))

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Now consider the question of whether multiplication with a negative number distributes over addition and subtraction of integers. For example, would (-10) \times 5 + (-10) \times (-3) also have the answer -20, as does (-10) \times (5 + (-3))?

11. What must (-10) \times (-3) be equal to, if we want (-10) \times 5 + (-10) \times (-3) to be equal to -20?

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In order to ensure that multiplication distributes over addition and subtraction in the system of integers, we have to agree that

(a negative number) \times (a negative number) is a positive number,

for example (-10) \times (-3) = 30.

12. Calculate.

(a) (-10) \times (-5)

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(b) (-10) \times 5

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(c) 10 \times 5

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(d) 10 \times (-5)

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(e) (-20) \times (-10) + (-20) \times (-6) (f) (-20) \times ((-10) + (-6))

(g) (-20) \times (-10) - (-20) \times (-6) (h) (-20) \times ((-10) - (-6))

Here is a summary of the properties of integers that make it possible to do calculations with integers:

division with integers

1. (a) Calculate 25 \times 8.

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(b) How much is 200 \div 25?

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(c) How much is 200 \div 8?

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2. Calculate.

(a) 25 \times (-8) (b) (-125) \times 8

3. Use the work you have done for question 2 to write the answers for the following division questions:

(a) (-1 000) \div (-125)

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(b) (-1 000) \div 8

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(c) (-200) \div 25

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(d) (-200) \div 8

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4. Can you also work out the answers for the following division questions by using the work you have done for question 2?

(a) 1 000 \div (-125)

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(b) (-1 000) \div (-8)

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(c) (-100) \div (-25)

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(d) 100 \div (-25)

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When two numbers are multiplied, for example 30 \times 4 = 120, the word "product" can be used in various ways to describe the situation:

• • An expression that specifies multiplication only, such as 30 \times 4, is called a product or a product expression.
• • The answer obtained is also called the product of the two numbers. For example, 120 is called the product of 30 and 4.

An expression that specifies division only, such as 30 \div 5, is called a quotient or a quotient expression. The answer obtained is also called the quotient of the two numbers. For example, 6 is called the quotient of 30 and 5.

5. In each case, state whether you agree or disagree with the statement, and give an example to illustrate your answer.

(a) The quotient of a positive and a negative integer is negative.

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(b) The quotient of a positive and a positive integer is negative.

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(c) The quotient of a negative and a negative integer is negative.

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(d) The quotient of a negative and a negative integer is positive.

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6. Do the necessary calculations to enable you to provide the values of the quotients.

(a) (-500) \div (-20) (b) (-144) \div 6

(c) 1 440 \div (-60) (d) (-1 440) \div (-6)

(e) -14 400 \div 600 (f) 500 \div (-20)

the associative properties of operations with integers

Multiplication of whole numbers is associative. This means that in a product with several factors, the factors can be placed in any sequence, and the calculations can be performed in any sequence. For example, the following sequences of calculations will all produce the same answer:

A. 2 \times 3, the answer of 2 \times 3 multiplied by 5, the new answer multiplied by 10

B. 2 \times 5, the answer of 2 \times 5 multiplied by 10, the new answer multiplied by 3

C. 10 \times 5, the answer of 10 \times 5 multiplied by 3, the new answer multiplied by 2

D. 3 \times 5, the answer of 3 \times 5 multiplied by 2, the new answer multiplied by 10

1. Do the four sets of calculations given in A to D to check whether they really produce the same answers.

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D. 15 \times 2 = 30; 30 \times 10 = 30000

2. (a) If the numbers 3 and 10 in the calculation sequences A, B, C and D are replaced with -3 and -10, do you think the four answers will still be the same?

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(b) Investigate, to check your expectation.

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The calculation sequence A can be represented in symbols in only two ways:

• • 2 \times 3 \times 5 \times 10. The convention to work from left to right unless otherwise indicated with brackets ensures that this representation corresponds to A.
• • 5 \times (2 \times 3) \times 10, where brackets are used to indicate that 2 \times 3 should be calculated first. When brackets are used, there are different possibilities to describe the same sequence.

3. Express the calculation sequences B, C and D given on page 44 symbolically, without using brackets.

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4. Investigate, in the same way that you did for multiplication in question 2, whether addition with integers is associative. Use sequences of four integers.

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5. (a) Calculate: 80 - 30 + 40 - 20

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(b) Calculate: 80 + (-30) + 40 + (-20)

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(c) Calculate: 30 - 80 + 20 - 40

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(d) Calculate: (-30) + 80 + (- 20) + 40

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(e) Calculate: 20 + 30 - 40 - 80

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mixed calculations with integers

1. Calculate.

(a) -3 \times 4 + (-7) \times 9 (b) -20(-4 - 7)

(c) 20 \times (-5) - 30 \times 7 (d) -9(20 - 15)

(e) -8 \times (-6) - 8 \times 3 (f) (-26 - 13) \div (-3)

(g) -15 \times (-2) + (-15) \div (-3) (h) -15(2 - 3)

(i) (-5 + -3) \times 7 (j) -5 \times (-3 + 7) + 20 \div (-4)

2. Calculate.

(a) 20 \times (-15 + 6) - 5 \times (-2 - 8) - 3 \times (-3 - 8)

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(b) 40 \times (7 + 12 - 9) + 25 \div (-5) - 5 \div 5

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(c) -50(20 - 25) + 30(-10 + 7) - 20(-16 + 12)

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(d) -5 \times (-3 + 12 - 9)

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(e) -4 \times (30 - 50) + 7 \times (40 - 70) - 10 \times (60 - 100)

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(f) -3 \times (-14 + 6) \times (-13 + 7) \times (-20 + 5)

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(g) 20 \times (-5) + 10 \times (-3) + (-5) \times (-6) - (3 \times 5)

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(h) -5(-20 - 5) + 10(-7 - 3) - 20(-15 - 5) + 30(-40 - 35)

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(i) (-50 + 15 - 75) \div (-11) + (6 - 30 + 12) \div (-6)

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2.4 Squares, cubes and roots with integers

squares and cubes of integers

1. Calculate.

(a) 20 \times 20

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(b) 20 \times (-20)

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2. Write the answers for each of the following:

(a) (-20) \times 20

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(b) (-20) \times (-20)

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3. Complete the table.

x	1	-1	2	-2	5	-5	10	-10
x2 which isx \times x	1	1	4	4	25	25	100	100
x3	1	-1	8	-8	125	-125	1 000	-1 000

4. In each case, state for which values of x, in the table in question 3, the given statement is true.

(a) x3 is a negative number

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(b) x2 is a negative number

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(c) x2 > x3

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(d) x2 < x3

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5. Complete the table.

x	3	-3	4	-4	6	-6	7	-7
x2	9	9	16	16	36	36	49	49
x3	27	-27	64	-64	216	-216	343	-343

6. Ben thinks of a number. He adds 5 to it, and his answer is 12.

(a) What number did he think of?

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(b) Is there another number that would also give 12 when 5 is added to it?

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7. Lebo also thinks of a number. She multiplies the number by itself and gets 25.

(a) What number did she think of?

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(b) Is there more than one number that will give 25 when multiplied by itself?

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8. Mary thinks of a number and calculates (the number) \times (the number) \times (the number). Her answer is 27.

What number did Mary think of?

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9. Write the positive square root and the negative square root of each number.

(a) 64

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(b) 9

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10. Complete the table.

Number	1	4	9	16	25	36	49	64
Positive square root	1	2	3	4	5	6	7	8
Negative square root	-1	-2	-3	-4	-5	-6	-7	-8

11. Complete the tables.

(a)	x	1	2	3	4	5	6	7	8
	x3	1	8	27	64	125	216	343	512

(b)	x	-1	-2	-3	-4	-5	-6	-7	-8
	x3	-1	-8	-27	-64	-125	-216	-343	-512

12. Complete the table.

Number	-1	8	-27	-64	-125	-216	1 000
Cube root	-1	2	-3	-4	-5	-6	10

The symbol is used to indicate "root".

represents the cube root of -125. That means = -5.

represents the positive square root of 36, and - represents the negative square

root. The "2" that indicates "square" is normally omitted, so = 6 and - = -6.

13. Complete the table.

			-			-
-2	11	-4	-8	8	-1	-1	-6

1. Use the numbers -8, -5 and -3 to demonstrate each of the following:

(a) Multiplication with integers distributes over addition.

-8(-5 + (-3)) = -8(-5) + (-8)(-3)

(b) Multiplication with integers distributes over subtraction.

-8(-5 - (-3)) = -8(-5) - (-8)(-3)

(c) Multiplication with integers is associative.

-8 \times (-3) \times (-5) = -5 \times (-8) \times (-3)

(d) Addition with integers is associative.

-8 + (-3) + (-5) = -5 + (-8) + (-3)

2. Calculate each of the following without using a calculator:

(a) 5 \times (-2)3 (b) 3 \times (-5)2

(c) 2 \times (-5)3 (d) 10 \times (-3)2

3. Use a calculator to calculate each of the following:

(a) 24 \times (-53) + (-27) \times (-34) - (-55) \times 76

= -1 272 + 918 + 4 180 = 3 826

(b) 64 \times (27 - 85) - 29 \times (-47 + 12)

= 64(-58) - 29(-35) = -3 712 + 1 015 = -2 697

4. Use a calculator to calculate each of the following:

(a) -24 \times 53 + 27 \times 34 + 55 \times 76

(b) 64 \times (-58) + 29 \times (47 - 12)

If you don't get the same answers in questions 3 and 4, you have made mistakes.

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In this chapter, you will revise work you have done on squares, cubes, square roots and cube roots. You will learn about laws of exponents that will enable you to do calculations using numbers written in exponential form.

Very large numbers are written in scientific notation. Scientific notation is a convenient way of writing very large numbers as a product of a number between 1 and 10 and a power of 10.

3.1 Revision 53

3.2 Working with integers 58

3.3 Laws of exponents 60

3.4 Calculations 70

3.5 Squares, cubes and roots of rational numbers 71

3.6 Scientific notation 74

3 Exponents

3.1 Revision

exponential notation

1. Calculate.

(a) 2 \times 2 \times 2 (b) 2 \times 2 \times 2 \times 2 \times 2 \times 2

(c) 3 \times 3 \times 3 (d) 3 \times 3 \times 3 \times 3 \times 3 \times 3

2. Write each of the following in exponential form:

(a) 2 \times 2 \times 2 (b) 2 \times 2 \times 2 \times 2 \times 2 \times 2

(c) 3 \times 3 \times 3 (d) 3 \times 3 \times 3 \times 3 \times 3 \times 3

3. Calculate.

(a) 52 (b) 25

(c) 102 (d) 152

(e) 34 (f) 43

(g) 23 (h) 32

squares

1. Complete the table.

	Number	Square the number	Exponential form	Square
(a)	1	1 \times 1	12	1
(b)	2	2 \times 2	22	4
(c)	3	3 \times 3	32	9
(d)	4	4 \times 4	42	16
(e)	5	5 \times 5	52	25
(f)	6	6 \times 6	62	36
(g)	7	7 \times 7	72	49
(h)	8	8 \times 8	82	64
(i)	9	9 \times 9	92	81
(j)	10	10 \times 10	102	100
(k)	11	11 \times 11	112	121
(l)	12	12 \times 12	122	144

2. Calculate the following:

(a) 32 \times 42 (b) 22 \times 32

(c) 22 \times 52 (d) 22 \times 42

3. Complete the following statements to make them true:

(a) 32 \times 42 =

(b) 22 \times 32 =

(c) 22 \times 52 =

(d) 22 \times 42 =

cubes

1. Complete the table.

	Number	Cube the number	Exponential form	Cube
(a)	1	1 \times 1 \times 1	13	1
(b)	2	2 \times 2 \times 2	23	8
(c)	3	3 \times 3 \times 3	33	27
(d)	4	4 \times 4 \times 4	43	64
(e)	5	5 \times 5 \times 5	53	125
(f)	6	6 \times 6 \times 6	63	216
(g)	7	7 \times 7 \times 7	73	343
(h)	8	8 \times 8 \times 8	83	512
(i)	9	9 \times 9 \times 9	93	729
(j)	10	10 \times 10 \times 10	103	1 000

2. Calculate the following:

(a) 23 \times 33 (b) 23 \times 53

(c) 23 \times 43 (d) 13 \times 93

3. Which of the following statements are true? If a statement is false, rewrite it as a true statement.

(a) 23 \times 33 = 63 (b) 23 \times 53 = 73

(c) 23 \times 43 = 83 (d) 13 \times 93 = 103

square and cube roots

1. Complete the table.

	Number	Square of the number	Square root of the square of the number	Reason
(a)	1	1	1	1 \times 1 = 1
(b)	2	4	2	2 \times 2 = 4
(c)	3	9	3	3 \times 3 = 9
(d)	4	16	4	4 \times 4 = 16
(e)	5	25	5	5 \times 5 = 25
(f)	6	36	6	6 \times 6 = 36
(g)	7	49	7	7 \times 7 = 49
(h)	8	64	8	8 \times 8 = 64
(i)	9	81	9	9 \times 9 = 81
(j)	10	100	10	10 \times 10 = 100
(k)	11	121	11	11 \times 11 = 121
(l)	12	144	12	12 \times 12 = 144

2. Calculate the following. Justify your answer.

(a) (b)  

(c) (d)  

3. Complete the table.

	Number	Cube of the number	Cube root of the cube of the number	Reason
(a)	1	1	1	1 \times 1 \times 1 = 1
(b)	2	8	2	2 \times 2 \times 2 = 8
(c)	3	27	3	3 \times 3 \times 3 = 27
(d)	4	64	4	4 \times 4 \times 4 = 64
(e)	5	125	5	5 \times 5 \times 5 = 125
(f)	6	216	6	6 \times 6 \times 6 = 216
(g)	7	343	7	7 \times 7 \times 7 = 343
(h)	8	512	8	8 \times 8 \times 8 = 512
(i)	9	729	9	9 \times 9 \times 9 = 729
(j)	10	1 000	10	10 \times 10 \times 10 = 1 000

4. Calculate the following and give reasons for your answers:

(a) (b)

(c) (d)  

(e) (f)  

3.2 Working with integers

representing integers in exponential form

1. Calculate the following, without using a calculator:

(a) -2 \times -2 \times -2 (b) -2 \times -2 \times -2 \times -2

(c) -5 \times -5 (d) -5 \times -5 \times -5

(e) -1 \times -1 \times -1 \times -1 (f) -1 \times -1 \times -1

2. Calculate the following:

(a) -22 (b) (-2)2

(c) (-5)3 (d) -53

3. Use your calculator to calculate the answers to question 2.

(a) Are your answers to question 2(a) and (b) different or the same as those of the calculator?

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(b) If your answers are different to those of the calculator, try to explain how the calculator did the calculations differently from you.

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4. Write the following in exponential form:

(a) -2 \times -2 \times -2 (b) -2 \times -2 \times -2 \times -2

(c) -5 \times -5 (d) -5 \times -5 \times -5

(e) -1 \times -1 \times -1 \times -1 (f) -1 \times -1 \times -1

5. Calculate the following:

(a) (-3)2 (b) (-3)3

(c) (-2)4 (d) (-2)6

(e) (-2)5 (f) (-3)4

6. Say whether the sign of the answer is negative or positive. Explain why.

(a) (-3)6 (b) (-5)11

(c) (-4)20 (d) (-7)5

7. Say whether the following statements are true or false. If a statement is false rewrite it as a correct statement.

(a) (-3)2 = -9 (b) -32 = 9

(c) (-52) = -52 (d) (-1)3 = -13

(e) (-6)3 = -18 (f) (-2)6 = 26

3.3 Laws of exponents

product of powers

1. A product of 2s is given below. Describe it using exponential notation, that is, write it as a power of 2.

2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2

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2. Express each of the following as a product of the powers of 2, as indicated by the brackets.

(a) (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2)

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(b) (2 \times 2 \times 2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2 \times 2) \times (2 \times 2)

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(c) (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2)

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(d) (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2)

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(e) (2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2) \times (2 \times 2)

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3. Complete the following statements so that they are true. You may want to refer to your answers to question 2 (a) to (e) to help you.

(a) 23 \times

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= 212

(b) 25 \times

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\times 22 =212

(c) 22 \times 22 \times 22 \times 22 \times 22 \times 22 =

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(d) 28 \times

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= 212

(e) 23 \times 23 \times 23 \times

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= 212

(f) 26 \times

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= 212

(g) 22\times 210 =

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Suppose we are asked to simplify: 32 \times 34.

The solution is: 32 \times 34 = 9 \times 81

= 729

= 36

We can explain this solution in the following manner:

32 \times 34 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 36

2 factors 4 factors 6 factors

4. Complete the table.

	Product of powers	Repeated factor	Total number of times the factor is repeated	Simplified form
(a)	27 \times 23	2	10	210
(b)	52\times 54	5	6	56
(c)	41\times 45	4	6	46
(d)	63\times 62	6	5	65
(e)	28 \times 22	2	10	210
(f)	53 \times 53	5	6	56
(g)	42 \times 44	4	6	46
(h)	21 \times 29	2	10	210

5. What is wrong with these statements? Correct each one.

(a) 23 \times 24 = 212 (b) 10 \times 102 \times 103 = 101 \times 2 \times 3 = 106

(c) 32 \times 33 = 36 (d) 53 \times 52 = 15 \times 10

6. Express each of the following numbers as a single power of 10.

Example: 1 000 000 as a power of 10 is 106.

(a) 100 (b) 1 000 (c) 10 000

(d) 102 \times 103 \times 104 (e) 100 \times 1 000 \times 10 000 (f) 1 000 000 000

7. Write each of the following products in exponential form:

(a) x \times x \times x \times x \times x \times x \times x \times x \times x =

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(b) (x \times x) \times (x \times x \times x) \times (x \times x \times x \times x)

= x2 \times x3 \times x4 = x2 + 3 + 4 = x9

(c) (x \times x \times x \times x) \times (x \times x) \times (x \times x) \times x

= x4 \times x2 \times x2 \times x = x4 + 2 + 2 + 1 = x9

(d) (x \times x \times x \times x \times x \times x) \times (x \times x \times x)

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(e) (x \times x \times x) \times (y \times y \times y)

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(f) (a \times a) \times (b \times b)

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8. Complete the table.

	Product of powers	Repeated factor	Total number of times the factor is repeated	Simplified form
(a)	x7 \times x3	x	10	x10
(b)	x2 \times x4	x	6	x6
(c)	x1 \times x5	x	6	x6
(d)	x3 \times x2	x	5	x5
(e)	x8 \times x2	x	10	x10
(f)	x3 \times x3	x	6	x6
(g)	x1 \times x9	x	10	x10

raising a power to a power

1. Complete the table of powers of 2.

x	1	2	3	4	5	6	7	8	9	10	11
2x	2	4	8	16	32	64	128	256	512	1 024	2 048
	21	22	23	24	25	26	27	28	29	210	211

x	12	13	14	15	16	17	18
2x	4 096	8 192	16 384	32 768	65 536	131 072	262 144
	212	213	214	215	216	217	218

2. Complete the table of powers of 3.

x	1	2	3	4	5	6	7	8	9
3x	3	9	27	81	243	729	2 187	6 561	19 683
	31	32	33	34	35	36	37	38	39

x	10	11	12	13	14
3x	59 049	177 147	531 441	1 594 323	4 782 969
	310	311	312	313	314

3. Complete the table. You can read the values from the tables you made in questions 1 and 2.

Product of powers	Repeated factor	Power of powernotation	Total number of repetitions	Simplified form	Value
24 \times 24 \times 24	2	(24)3	12	212	4 096
32 \times 32 \times 32 \times 32	3	(32)4	8	38	6 561
23 \times 23 \times 23 \times 23 \times 23	2	(23)5	15	215	32 768
34 \times 34 \times 34	3	(34)3	12	312	531 441
26 \times 26 \times 26	2	(26)3	18	218	262 144

4. Use your table of powers of 2 to find the answers for the following:

(a) 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 =

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=

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(b) (2 \times 2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) =

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=

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(c) 163 =

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=

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=

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5. Use your table of powers of 2 to find the answers for the following:

(a) Is 163 = 212?

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(b) Is 24 \times 24 \times 24 = 212

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(c) Is 24 \times 23 = 212?

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(d) Is (24)3 = 24 \times 24 \times 24?

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(e) Is (24)3 = 212?

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(f) Is (24)3 = 24 + 3?

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(g) Is (24)3 = 24 \times 3?

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(h) Is (22)5 = 22 + 5?

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6. (a) Express 85 as a power of 2. It may help to first express 8 as a power of 2.

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(b) Can (23) \times (23) \times (23) \times (23) \times (23) be expressed as (23)5?

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(c) Is (23)5 = 23 + 5 or is (23)5 = 23 \times 5?

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7. (a) Express 43 as a power of 2.

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(b) Calculate 22 \times 22 \times 22 and express your answer as a single power of 2.

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(c) Can (22) \times (22) \times (22) be expressed as (22)3?

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(d) Is (22)3 = 22 + 3 or is (22)3 = 22 \times 3?

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8. Simplify the following.

Example: (102)2 = 102 \times 102 = 102 + 2 = 104 = 10 000

(a) (33)2

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(b) (43)2

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(c) (24)2

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(d) (92)2

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(e) (33)3

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(f) (43)3

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(g) (54)3

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(h) (92)3

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9. Simplify.

(a) (54)10

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(b) (104)5

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(c) (64)4

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(d) (54)10

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10. Write 512 as a power of powers of 5 in two different ways.

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To simplify (x2)5 we can write it out as a product of powers or we can use a shortcut.

(x2)5 = x2 \times x2 \times x2 \times x2 \times x2

= x \times x \times x \times x \times x \times x \times x \times x \times x \times x = x10

2 \times 5 factors = 10 factors

11. Complete the table.

	Expression	Write as a product of the powers and then simplify	Use the rule (am)n to simplify
(a)	(a4)5	a4 \times a4 \times a4 \times a4 \timesa4 = a4 + 4 + 4 + 4 + 4 = a20	(a4)5 = a4 \times 5 = a20
(b)	(b10)5	b10 \times b10 \times b10 \times b10 \times b10 = b10 + 10 + 10 + 10 +10 = b50	(b10)5 = b10 \times 5 = b50
(c)	(x7)3	x7 \times x7 \times x7 = x7 + 7 + 7 = x21	(x7)3 = x7 \times 3 = x21
(d)	(s6)4	s6 \times s6 \times s6 \times s6 = s6 + 6 + 6 + 6 = s24	(s6)4 = s6 \times 4 = s24
(e)	(y3)7	y3 \times y3 \times y3 \times y3 \times y3 \times y3 \times y3 = y3+ 3 + 3 + 3 + 3 + 3 + 3 + 3 = y21	y3 \times 7 = y21

power of a product

1. Complete the table. You may use your calculator when you are not sure of a value.

	x	1	2	3	4	5
(a)	2x	21 = 2	22 = 4	23 = 8	24 = 16	25 = 32
(b)	3x	31 = 3	32 = 9	33 = 27	34 = 81	35 = 243
(c)	6x	61 = 6	62 = 36	63 = 216	64 = 1 296	65 = 6 480

2. Use the table in question 1 to answer the questions below. Are these statements true or false? If a statement is false rewrite it as a correct statement.

(a) 62 = 22 \times 32 (b) 63 = 23 \times 33

(c) 65 = 25 \times 35 (d) 68 = 24 \times 34

3. Complete the table.

	Expression	The bases of the expression are factors of …	Equivalent expression
(a)	26 \times 56	10	106
(b)	32 \times 42	12	122
(c)	42 \times 22	8	82
(d)	75 \times 85	56	565
(e)	23 \times 153	30	303
(f)	35 \times x5	3x	(3x)5
(g)	72 \times z2	7z	(7z)2
(h)	43 \times y3	4y	(4y)3
(i)		2m	(2m)6
(j)		2m	(2m)3
(k)	210 \times y10	2y	(2y)10

122 can be written in terms of its factors as (2 \times 6)2 or as (3 \times 4)2.

We already know that 122 = 144.

What this tells us is that both (2 \times 6)2 and (3 \times 4)2 also equal 144.

We write 122 = (2 \times 6)2 or 122 = (3 \times 4)2

= 22 \times 62 = 32 \times 42

= 4 \times 36 = 9 \times 16

= 144 = 144

4. Write each of the following expressions as an expression with one base:

Example: 310 \times 210 = (3 \times 2)10 = 610

(a) 32 \times 52 (b) 53 \times 23 (c) 74 \times 44

(d) 23 \times 63 (e) 44 \times 24 (f) 52 \times 72

5. Write the following as a product of powers:

Example: (3x)3 = 33 \times x3 = 27x3

(a) 63 (b) 152 (c) 214

(d) 65 (e) 182 (f) (st)7

(g) (ab)3 (h) (2x)2 (i) (3y)5

(j) (3c)2 (k) (gh)4 (l) (4x)3

6. Simplify the following expressions:

Example: 32 \times m2 = 9 \times m2 = 9m2

(a) 35 \times b5 (b) 26 \times y6 (c) x2 \times y2

(d) 104 \times x4 (e) 33 \times x3 (f) 52 \times t2

(g) 63 \times m7 (h) 122 \times a2 (i) n3 \times p9

a quotient of powers

Consider the following table:

x	1	2	3	4	5	6
2x	2	4	8	16	32	64
3x	3	9	27	81	243	729
5x	5	25	125	625	3 125	15 625

Answer questions 1 to 4 by referring to the table when you need to.

1. Give the value of each of the following:

(a) 34 (b) 25 (c) 56

2. (a) Calculate 36 \div 33 (Read the values of 36 and 33 from the table and then divide.

You may use a calculator where necessary.)

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(b) Calculate 36- 3

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(c) Is 36 \div 33 equal to 33? Explain.

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3. (a) Calculate the value of 26- 2 (b) Calculate the value of 26 \div 22

(c) Calculate the value of 26\div 2 (d) Read from the table the value of 23

(e) Read from the table the value of 24

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(f) Which of the statements below is true? Give an explanation for your answer.

A. 26 \div 22 = 26 - 2 = 24 B. 26 \div 22 = 26 \div 2 = 23

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4. Say which of the statements below are true and which are false. If a statement is false rewrite it as a correct statement.

(a) 56 \div 54 = 56 \div 4 (b) 34 - 1 = 34 \div 3

(c) 56 \div 5 = 56 - 1 (d) 25 \div 23 = 22

5. Simplify the following. Do not use a calculator.

Example: 317 \div 312 = 317 - 12 = 35 = 243

(a) 212 \div 210 (b) 617 \div 614

(c) 1020 \div 1014 (d) 511 \div 58

6. Simplify:

(a) x12 \div x10 (b) y17 \div y14

(c) t20 \div t14 (d) n11 \div n8

the power of zero

1. Simplify the following:

(a) 212 \div 212 (b) 617 \div 617

(c) 614 \div 614 (d) 210 \div 210

2. Simplify the following:

(a) 1000 (b) x0 (c) (100x)0 (d) (5x3)0

 

3.4 Calculations

mixed operations

Simplify the following:

1. 33 + \times 2 2. 5 \times (2 + 3)2 + (-1)0

3. 32 \times 23 + 5 \times 4. + (4 - 1)2

5. \times + + 32 \times 10 6. 43 \div 23 +

 

3.5 Squares, cubes and roots of rational numbers

squaring a fraction

1. Complete the table.

	Fraction	Square the fraction	Value of the square of the fraction
(a)		\times	\times =
(b)
(c)
(d)
(e)
(f)
(g)
(h)

2. Calculate the following:

(a) (b) (c)  

3. (a) Use the fact that 0,6 can be written as to calculate (0,6)2.

(0,6)2 = = = 0,36

(b) Use the fact that 0,8 can be written as to calculate (0,8)2.

(0,8)2 = = = 0,64

finding the square root of a fraction

1. Complete the table.

	Fraction	Writing the fraction as a product of factors	Square root
(a)
(b)
(c)
(d)

2. Determine the following:

(a) (b) (c) (d)  

3. (a) Use the fact that 0,01 can be written as to calculate .

= =

(b) Use the fact that 0,49 can be written as to calculate .

= = = 0,7

4. Calculate the following:

(a) (b) (c)

cubing a fraction

1. Calculate the following:

(a) (b) (c) (d)

2. (a) Use the fact that 0,6 can be written as to find (0,6)3.

(0,6)3 = = = 0,216

(b) Use the fact that 0,8 can be written as to calculate (0,8)3.

(0,8)3 = = = 0,512

(c) Use the fact that 0,7 can be written as to calculate (0,7)3.

(0,8)3 = = = 0,512

3.6 Scientific notation

very large numbers

1. Express each of the following as a single number. Do not use a calculator.

Example: 7,56 \times 100 can be written as 756.

(a) 3,45 \times 100 (b) 3,45 \times 10 (c) 3,45 \times 1 000

(d) 2,34 \times 102 (e) 2,34 \times 10 (f) 2,34 \times 103

(g) 104 \times 102 (h) 100 \times 106 (i) 3,4 \times 105

2. Write the following numbers in scientific notation:

(a) 367 000 000 (b) 21 900 000

(c) 600 000 000 000 (d) 178

3. Write each of the following numbers in the ordinary way.

For example: 3,4 \times 105 written in the ordinary way is 340 000.

(a) 1,24 \times 108 (b) 9,2074 \times 104

(c) 1,04 \times 106 (d) 2,05 \times 103

4. The age of the universe is 15 000 000 000 years. Express the age of the universe in scientific notation.

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5. The average distance from the Earth to the Sun is 149 600 000 km. Express this distance in scientific notation.

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6. Explain why the number 24 \times 103 is not in scientific notation.

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7. Calculate the following. Do not use a calculator.

Example: 3 000 000 \times 90 000 000 = 3 \times 106 \times 9 \times 107 = 3 \times 9 \times 106 + 7

= 27 \times 1013 = 270 000 000 000 000

(a) 13 000 \times 150 000 (b) 200 \times 6 000 000

(c) 120 000 \times 120 000 000 (d) 2,5 \times 40 000 000

8. Use > or < to compare these numbers:

(a) 1,3 \times 109

2,4 \times 107 (b) 6,9 \times 102

4,5 \times 103

(c) 7,3 \times 104

7,3 \times 102 (d) 3,9 \times 106

3,7 \times 107

1. Calculate:

(a) 112

(b) 32 \times 42

(c) 63

(d)

(e) (-3)2

(f)

2. Simplify:

(a) 34 \times m6

(b) b2 \times n6

(c) y12 \div y5

(d) (102)3

(e) (2w2)3

(f) (3d5)(2d)3

3. Calculate:

(a)

(b)

(c) (64y 2)0

(d) (0,7)2

4. Simplify:

(a) (22 + 4)2 + (b) - 5 \times 32

5. Write 3 \times 109 in the ordinary way.

6. The first birds appeared on Earth about 208 000 000 years ago. Write this number in scientific notation.

2,08 \times 106